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A008919 Numbers n such that n written backwards is a nontrivial multiple of n. 11
1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, 10891089, 10999989, 21782178, 21999978, 108901089, 109999989, 217802178, 219999978, 1089001089, 1098910989, 1099999989, 2178002178, 2197821978, 2199999978, 10890001089 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

There are 2*Fibonacci([(n-2)/2]) terms with n digits (this is essentially twice A103609). - N. J. A. Sloane, Mar 20 2013

REFERENCES

W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays (1939, page 13); 13th ed. New York: Dover, pp. 14-15, 1987.

Gardiner, Anthony, and A. D. Gardiner. Discovering mathematics: The art of investigation. Oxford University Press, 1987.

G. H. Hardy, A Mathematician's Apology (Cambridge Univ. Press, 1940, reprinted 2000), pp. 104-105 (describes this problem as having "nothing in [it] which appeals much to a mathematician.").

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Martin Beech, A Computer Conjecture of a Non-Serious Theorem, Mathematical Gazette, 74 (No. 467, March 1990), 50-51.

Patrick De Geest, Palindromic Products of Integers and their Reversals

D. J. Hoey, Palintiples

D. J. Hoey, Palintiples [Cached copy]

Benjamin V. Holt, Some General Results and Open Questions on Palintiple Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 14, Paper A42, 2014.

Benjamin V. Holt, A Determination of Symmetric Palintiples, arXiv:1410.2356 [math.NT], 2014.

Benjamin V. Holt, Families of Asymmetric Palintiples Constructed from Symmetric and Shifted-Symmetric Palintiples, arXiv:1412.0231 [math.NT], 2014.

Benjamin V. Holt, Derived Palintiple Families and Their Palinomials, arXiv preprint, 2015.

L. H. Kendrick, Young Graphs: 1089 et al, arXiv:1410.0106 [math.NT], 2014.

Leonard F. Klosinski and Dennis C. Smolarski, On the Reversing of Digits, Math. Mag., 42 (1969), 208-210.

Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129-132.

N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.

R. Webster and G. Williams, On the Trail of Reverse Divisors: 1089 and All that Follow, Mathematical Spectrum, Applied Probability Trust, Sheffield, Vol. 45, No. 3, 2012/2013, pp. 96 - 102.

Eric Weisstein's World of Mathematics, Reversal

FORMULA

If reverse(n) = k*n in base 10, then k = 1, 4 or 9 [Klosinski and Smolarski]. Hence A008919 is the union of A001232 and A008918. - David W. Wilson

MAPLE

P:=proc(q) local a, b, n; for n from 1 to q do a:=n; b:=0;

while a>0 do b:=b*10+(a mod 10); a:=trunc(a/10); od;

if type(b/n, integer) then if b/n>1 then print(n); fi; fi;

od; end: P(10^10); # Paolo P. Lava, Jul 29 2014

MATHEMATICA

Reap[ Do[ If[ Reverse[ IntegerDigits[n]] == IntegerDigits[4*n], Print[n]; Sow[n]]; If[ Reverse[ IntegerDigits[n + 11]] == IntegerDigits[9*(n + 11)], Print[n + 11]; Sow[n + 11]], {n, 78, 2*10^10, 100}]][[2, 1]] (* Jean-Fran├žois Alcover, Jun 19 2012, after David W. Wilson, assuming n congruent to 78 or 89 mod 100 *)

okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[ Flatten[ {99#, 198#}&/@Flatten[Table[FromDigits/@Select[Tuples[ {0, 1}, n], okQ], {n, 10}]]]] (* Harvey P. Dale, Jul 03 2013 *)

PROG

(Haskell)

a008919 n = a008919_list !! (n-1)

a008919_list = [x | x <- [1..],

                    let (x', m) = divMod (a004086 x) x, m == 0, x' > 1]

-- Reinhard Zumkeller, Feb 03 2012

CROSSREFS

Cf. A001232, A004086, A008918, A214927, A103609. Reversals are in A031877.

Sequence in context: A175698 A110819 A071685 * A110843 A023093 A001232

Adjacent sequences:  A008916 A008917 A008918 * A008920 A008921 A008922

KEYWORD

nonn,base,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Corrected and extended by David W. Wilson Aug 15 1996, Dec 15 1997

STATUS

approved

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Last modified August 31 02:53 EDT 2015. Contains 261228 sequences.